Learning in a Small/Big World

Savage (1972) lays down the foundation of Bayesian decision theory, but asserts that it is not applicable in big worlds where the environment is complex. Using the theory of finite automaton to model belief formation, this paper studies the characteristics of optimal learning behavior in small and big worlds, where the complexity of the environment is low and high, respectively, relative to the cognitive ability of the decision maker. Confirming Savage's claim, optimal learning behavior is closed to Bayesian in small worlds but significantly different in big worlds. In addition, I show that in big worlds, the optimal learning behavior could exhibit a wide range of well-documented non-Bayesian learning behavior, including the use of heuristic, correlation neglect, persistent over-confidence, inattentive learning, and other behaviors of model simplification or misspecification. These results establish a clear and testable relationship between the prominence of non-Bayesian learning behavior, complexity and cognitive ability.